On Lipschitz implicit function theorems in Banach spaces and applications

Abstract

This paper presents several Lipschitz implicit function theorems for maps between Banach spaces. The results use semi-inner products as a means for understanding the geometry of normed spaces. The arguments rely on corresponding geometric properties of operators on normed spaces, and on Ekeland's variational principle, in contrast with classic results such as the inverse function theorems of Nash and Moser and their use of Newton iteration procedures. The results are illustrated with an application to a central problem in mathematical economics, characterizing Lipschitz behavior of Pareto optimal allocations in a model of trade over an infinite horizon.